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24.2.1 Energy Equation

The enthalpy of the material is computed as the sum of the sensible enthalpy, $h$, and the latent heat, $\Delta H$:

 H = h + \Delta H (24.2-1)


 h = h_{\rm ref} + \int_{T_{\rm ref}}^T c_p dT (24.2-2)

and $h_{\rm ref}$ = reference enthalpy
  $T_{\rm ref}$ = reference temperature
  $c_p$ = specific heat at constant pressure

The liquid fraction, $\beta$, can be defined as

$\displaystyle \beta = 0$ $\textstyle \mbox{if}$ $\displaystyle T < T_{\rm solidus}$  
$\displaystyle \beta = 1$ $\textstyle \mbox{if}$ $\displaystyle T > T_{\rm liquidus}$  
$\displaystyle \beta = \frac{T-T_{\rm solidus}}{T_{\rm liquidus} - T_{\rm solidus}}$ $\textstyle \mbox{if}$ $\displaystyle T_{\rm solidus} < T < T_{\rm liquidus}$ (24.2-3)

Equation  24.2-3 is referred to as the lever rule .

The latent heat content can now be written in terms of the latent heat of the material, $L$:

 \Delta H = \beta L (24.2-4)

The latent heat content can vary between zero (for a solid) and $L$ (for a liquid).

In the case of multicomponent solidification with species segregation; i.e., solidification or melting with species transport, the solidus and liquidus temperatures are computed instead of specified (Equations  24.2-5 and 24.2-6).

$\displaystyle T_{\rm solidus}$ $\textstyle =$ $\displaystyle T_{\rm melt} + \sum_{\rm solutes}{ K_i m_i Y_i }$ (24.2-5)
$\displaystyle T_{\rm liquidus}$ $\textstyle =$ $\displaystyle T_{\rm melt} + \sum_{\rm solutes}{ m_i Y_i }$ (24.2-6)

where $K_i$ is the partition coefficient of solute $i$, which is the ratio of the concentration in solid to that in liquid at the interface, $Y_i$ is the mass fraction of solute $i$, and $m_i$ is the slope of the liquidus surface with respect to $Y_i$. It is assumed that the last species material of the mixture is the solvent and that the other species are the solutes.

For solidification/melting problems, the energy equation is written as

 \frac{\partial}{\partial t} (\rho H) + \nabla \cdot (\rho {\vec v} H) = \nabla \cdot (k \nabla T) + S (24.2-7)

where $H$ = enthalpy (see Equation  24.2-1)
  $\rho$ = density
  $\vec{v}$ = fluid velocity
  $S$ = source term

The solution for temperature is essentially an iteration between the energy equation (Equation  24.2-7) and the liquid fraction equation (Equation  24.2-3). Directly using Equation  24.2-3 to update the liquid fraction usually results in poor convergence of the energy equation. In FLUENT, the method suggested by Voller and Swaminathan [ 386] is used to update the liquid fraction. For pure metals, where $T_{\rm solidus}$ and $T_{\rm liquidus}$ are equal, a method based on specific heat, given by Voller and Prakash [ 385], is used instead.

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